We have modelled the inner surface brightness profiles of 39 alleged 'core' galaxies with the core-Sersic model, and provide new physical parameters for the largest ever sample of 'core' galaxies fit with this model. When present, additional nuclear components were simultaneously modelled and the typical rms scatter of the fits (out to ~10 arcsec) is 0.02 mag/arcsec^2. Model-independent estimates of each core's break radius are shown to agree with those from the core-Sersic model, and a comparison with the Nuker model is provided. We found an absence of cores in what amounts to 18% of the sample which are reclassified here as Sersic galaxies with low values of n (< ~ 4) and thus shallow inner profile slopes. In general, galaxies with n<3 and sigma < 183 km/s do not have depleted cores. We derive updated relations between core-Sersic break radii, their associated surface brightness, bulge luminosity, central velocity dispersion, and predicted black hole mass for galaxies with depleted cores. With the possible exception of NGC 584, we confirm that the inner negative logarithmic profile slopes gamma are < ~ 0.3 for the 'core' galaxies, and 0 > gamma > -0.1 for six of these. Finally, the central stellar mass deficits are found to have values typically within a factor of 4 of the expected central black hole mass.